La conjecture de Birch et Swinnerton-Dyer 𝐩-adique
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 919, pp. 251-319.

La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre r du zéro en s=1 de la fonction L d’une courbe elliptique E définie sur 𝐐 est égal au rang r du groupe de ses points rationnels. On sait démontrer cette conjecture si r =0 ou 1, mais on n’a aucun résultat reliant r et r si r 2. Nous expliquerons comment Kato démontre que la fonction L p-adique attachée à E a, en s=1, un zéro d’ordre supérieur ou égal à r.

The classical Birch and Swinnerton-Dyer’s conjecture asserts that the order r of the zero at s=1 of the L-function of an elliptic curve E defined over 𝐐 is equal to the rank r of its group of rational points. This is a theorem if r =0 or 1, but there is no result relating r and r if r 2. We will explain how Kato proves that the p-adic L function attached to E has, at s=1, a zero of order at least r.

Classification : 11-02, 11F11, 11F67, 11F80, 11F85, 11G05, 11G16, 11G40, 11R33, 11R39, 11R56, 11S80, 11S99, 14F30, 14
Mot clés : courbe elliptique, fonction $L$ $p$-adique
Keywords: elliptic curve, $p$-adic $L$ function
@incollection{SB_2002-2003__45__251_0,
     author = {Colmez, Pierre},
     title = {La conjecture de {Birch} et {Swinnerton-Dyer} $\mathbf {p}$-adique},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     series = {Ast\'erisque},
     note = {talk:919},
     pages = {251--319},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     mrnumber = {2111647},
     zbl = {1094.11025},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__251_0/}
}
TY  - CHAP
AU  - Colmez, Pierre
TI  - La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique
BT  - Séminaire Bourbaki : volume 2002/2003, exposés 909-923
AU  - Collectif
T3  - Astérisque
N1  - talk:919
PY  - 2004
SP  - 251
EP  - 319
IS  - 294
PB  - Association des amis de Nicolas Bourbaki, Société mathématique de France
PP  - Paris
UR  - http://www.numdam.org/item/SB_2002-2003__45__251_0/
LA  - fr
ID  - SB_2002-2003__45__251_0
ER  - 
%0 Book Section
%A Colmez, Pierre
%T La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique
%B Séminaire Bourbaki : volume 2002/2003, exposés 909-923
%A Collectif
%S Astérisque
%Z talk:919
%D 2004
%P 251-319
%N 294
%I Association des amis de Nicolas Bourbaki, Société mathématique de France
%C Paris
%U http://www.numdam.org/item/SB_2002-2003__45__251_0/
%G fr
%F SB_2002-2003__45__251_0
Colmez, Pierre. La conjecture de Birch et Swinnerton-Dyer $\mathbf {p}$-adique, dans Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 919, pp. 251-319. http://www.numdam.org/item/SB_2002-2003__45__251_0/

[1] A. Agboola & B. Howard - “Anticyclotomic Iwasawa theory of CM elliptic curves”, preprint, 2003. | Numdam | MR | Zbl

[2] Y. Amice - “Interpolation p-adique”, Bull. Soc. math. France 92 (1964), p. 117-180. | Numdam | MR | Zbl

[3] -, “Duals” 1978), Nijmegen, Math. Institut Katholische Univ., 1978, p. 1-15.

[4] Y. Amice & J. Vélu - “Distributions p-adiques associées aux séries de Hecke”, in Journées arithmétiques de Bordeaux, Astérisque, vol. 24-25, Société Mathématique de France, 1975, p. 119-131. | Numdam | MR | Zbl

[5] K. Barré-Sirieix, G. Diaz, F. Gramain & G. Philibert - “Une preuve de la conjecture de Mahler-Manin”, Invent. Math. 124 (1996), p. 1-9. | MR | Zbl

[6] D. Barsky - “Fonctions zêta p-adiques d’une classe de rayon des corps totalement réels”, Groupe d'études d'analyse ultramétrique, 1977-1978 ; errata 1978-1979. | Numdam | Zbl

[7] A. Beilinson - “Higher regulators and values of L-functions”, J. Soviet Math. 30 (1985), p. 2036-2070. | MR | Zbl

[8] -, “Higher regulators of modular curves”, Contemp. Math. 55 (1986), p. 1-34. | MR

[9] J. Bellaïche - “Congruences endoscopiques et représentations galoisiennes”, Thèse, Université Paris 11, 2002.

[10] J. Bellaïche & G. Chenevier - “Formes non tempérées pour U(3) et conjectures de Bloch-Kato”, Ann. scient. Éc. Norm. Sup. 4 e série (à paraître). | Zbl

[11] D. Benois - “On Iwasawa theory of crystalline representations”, Duke Math. J. 104 (2000), p. 211-267. | MR | Zbl

[12] L. Berger - “Représentations p-adiques et équations différentielles”, Invent. Math. 148 (2002), p. 219-284. | MR | Zbl

[13] -, “Représentations de de Rham et normes universelles”, Bull. Soc. math. France (à paraître). | Numdam | Zbl

[14] M. Bertolini & H. Darmon - “Heegner points on Mumford-Tate curves”, Invent. Math. 126 (1996), p. 413-456. | MR | Zbl

[15] -, “A rigid analytic Gross-Zagier formula and arithmetic applications, with Appendix by B. Edixhoven”, Ann. of Math. 146 (1997), p. 117-147. | MR | Zbl

[16] -, “Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformisation”, Invent. Math. 131 (1998), p. 453-491. | MR | Zbl

[17] -, p-adic periods, p-adic L-functions and the p-adic uniformisation of Shimura curves”, Duke Math. J. 98 (1999), p. 305-334. | MR | Zbl

[18] -, “The p-adic L-functions of modular elliptic curves”, in 2001 and Beyond, Springer-Verlag, 2001. | MR

[19] -, “Iwasawa's main conjecture for elliptic curves in the anticyclotomic setting”, preprint.

[20] M. Bertolini, H. Darmon, A. Iovita & M. Spiess - “Teitelbaum's conjecture in the anticyclotomic setting”, Amer. J. Math. 124 (2002), p. 411-449. | MR | Zbl

[21] D. Bertrand - “Relations d'orthogonalité sur les groupes de Mordell-Weil”, in Séminaire de théorie des nombres, Paris 1984-85, Progress in Math., vol. 63, Birkhäuser, 1986, p. 33-39. | MR | Zbl

[22] A. Besser - “Syntomic regulators and p-adic integration I : rigid syntomic regulators”, Israel J. Math. 120 (2000), p. 291-334. | MR | Zbl

[23] -, “Syntomic regulators and p-adic integration II : K 2 of curves”, Israel J. Math. 120 (2000), p. 335-359. | MR | Zbl

[24] -, “The p-adic height pairings of Coleman-Gross and Nekovář”, in Proceedings of CNTA7, Montréal, à paraître.

[25] B. Birch & H. Swinnerton-Dyer - “Notes on elliptic curves. I”, J. reine angew. Math. 212 (1963), p. 7-25. | MR | Zbl

[26] -, “Notes on elliptic curves. II”, J. reine angew. Math. 218 (1965), p. 79-108. | MR | Zbl

[27] S. Bloch & K. Kato - “Tamagawa numbers of motives and L-functions”, in The Grothendieck Festschrift, vol. 1, Progress in Math., vol. 86, Birkäuser, 1990, p. 333-400. | MR | Zbl

[28] J.-B. Bost - “Algebraic leaves of algebraic foliations over number fields”, Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 161-221. | Numdam | MR | Zbl

[29] C. Breuil, B. Conrad, F. Diamond & R. Taylor - “On the modularity of elliptic curves over 𝐐 : wild 3-adic exercises”, J. Amer. Math. Soc. 14 (2001), p. 843-939. | MR | Zbl

[30] A. Brumer - “On the units of algebraic number fields”, Mathematika 14 (1967), p. 121-124. | MR | Zbl

[31] H. Carayol - “Sur les représentations l-adiques associées aux formes modulaires de Hilbert”, Ann. scient. Éc. Norm. Sup. 4 e série 19 (1986), p. 409-468. | Numdam | MR | Zbl

[32] J. Cassels - “Arithmetic on an elliptic curve”, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, p. 234-246. | MR | Zbl

[33] -, “Diophantine equations with special reference to elliptic curves”, J. London Math. Soc. (2) 41 (1966), p. 193-291. | MR | Zbl

[34] P. Cassou-Noguès - “Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques”, Invent. Math. 51 (1979), p. 29-59. | MR | Zbl

[35] A. Chambert-Loir - “Théorèmes d'algébricité en géométrie diophantienne (d'après J.-B. Bost, Y. André, D. et G. Chudnovsky)”, in Sém. Bourbaki 2000/01, Astérisque, vol. 282, Société Mathématique de France, 2002, exp. no 886, p. 175-209. | Numdam | MR | Zbl

[36] F. Cherbonnier & P. Colmez - “Représentations p-adiques surconvergentes”, Invent. Math. 133 (1998), p. 581-611. | MR | Zbl

[37] -, “Théorie d’Iwasawa des représentations p-adiques d’un corps local”, J. Amer. Math. Soc. 12 (1999), p. 241-268. | MR | Zbl

[38] D. Chudnovsky & G. Chudnovsky - “Padé approximations and Diophantine geometry”, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), p. 2212-2216. | MR | Zbl

[39] J. Coates - “The work of Mazur and Wiles on cyclotomic fields”, in Sém. Bourbaki 1980/81, Lect. Notes in Math., vol. 901, Springer, 1981, exp. no 575, p. 220-242. | Numdam | MR | Zbl

[40] -, “The work of Gross and Zagier on Heegner points and the derivatives of L-series”, in Sém. Bourbaki 1984/85, Astérisque, vol. 133-134, Société Mathématique de France, 1986, exp. no 635, p. 57-72. | Numdam | MR | Zbl

[41] J. Coates & A. Wiles - “On the conjecture of Birch and Swinnerton-Dyer”, Invent. Math. 39 (1977), p. 223-251. | MR | Zbl

[42] -, “On p-adic L-functions and elliptic units”, J. Austral. Math. Soc. Ser. A 26 (1978), p. 1-25. | MR

[43] R. Coleman - “Division values in local fields”, Invent. Math. 53 (1979), p. 91-116. | MR | Zbl

[44] -, “The dilogarithm and the norm residue symbol”, Bull. Soc. math. France 109 (1981), p. 373-402. | Numdam | MR | Zbl

[45] -, “A p-adic Shimura isomorphism and p-adic periods of modular forms”, Contemp. Math. 165 (1994), p. 21-51. | MR

[46] R. Coleman & B. Gross - p-adic heights on curves”, Adv. in Math. 17 (1989), p. 73-81. | MR | Zbl

[47] R. Coleman & A. Iovita - “Hidden structures on semi-stable curves”, preprint, 2003.

[48] P. Colmez - “Résidu en s=1 des fonctions zêta p-adiques”, Invent. Math. 91 (1988), p. 371-389. | MR | Zbl

[49] -, Intégration sur les variétés p-adiques, Astérisque, vol. 248, Société Mathématique de France, 1998. | Numdam | MR | Zbl

[50] -, “Représentations p-adiques d’un corps local”, in Proceedings of the International Congress of Mathematicians II (Berlin 1998), Doc. Mat. Extra, vol. II, Deutsche Math. Verein., 1998, p. 153-162.

[51] -, “Théorie d'Iwasawa des représentations de de Rham d'un corps local”, Ann. of Math. 148 (1998), p. 485-571. | MR | Zbl

[52] -, “Fonctions L p-adiques”, in Sém. Bourbaki 1998/99, Astérisque, vol. 266, Société Mathématique de France, 2000, exp. no 851, p. 21-58. | Numdam | MR

[53] -, “Arithmétique de la fonction zêta”, in La fonction zêta, Journées X-UPS, Éditions de l'École polytechnique, Palaiseau, 2002, p. 37-164.

[54] -, “Espaces de Banach de dimension finie”, J. Inst. Math. Jussieu 1 (2002), p. 331-439. | MR | Zbl

[55] -, “Invariants et dérivées de valeurs propres de frobenius”, preprint, 2003.

[56] -, “Les conjectures de monodromie p-adiques”, in Sém. Bourbaki 2001/02, Astérisque, vol. 290, Société Mathématique de France, 2003, exp. no 897, p. 53-101. | Numdam | MR | Zbl

[57] P. Colmez & J.-M. Fontaine - “Construction des représentations p-adiques semi-stables”, Invent. Math. 140 (2000), p. 1-43. | MR | Zbl

[58] C. Cornut - “Mazur's conjecture on higher Heegner points”, Invent. Math. 148 (2002), p. 495-523. | MR | Zbl

[59] H. Darmon - “Integration on p × and arithmetic applications”, Ann. of Math. 154 (2001), p. 589-639. | MR | Zbl

[60] H. Darmon & A. Iovita - “The anticyclotomic main conjecture for supersingular elliptic curves”, preprint, 2003. | Zbl

[61] D. Delbourgo - “On the p-adic Birch, Swinnerton-Dyer conjecture for non-semistable reduction”, J. Number Theory 95 (2002), p. 38-71. | MR | Zbl

[62] P. Deligne - “Formes modulaires et représentations -adiques”, in Sém. Bourbaki 1968/69, Lect. Notes in Math., vol. 179, Springer, 1971, exp. no 343, p. 139-172. | Numdam | Zbl

[63] -, “Valeurs de fonctions L et périodes d’intégrales”, in Automorphic forms, representations and L-functions, Proc. Symp. Pure Math., vol. 33, American Mathematical Society, 1979, p. 313-346. | Zbl

[64] -, “Preuve des conjectures de Tate et de Shafarevitch (d'après G. Faltings)”, in Sém. Bourbaki 1983/84, Astérisque, vol. 121-122, Société Mathématique de France, 1985, exp. no 616, p. 25-41. | Numdam | Zbl

[65] P. Deligne & K. Ribet - “Values of Abelian L-functions at Negative Integers Over Totally Real Fields”, Invent. Math. 59 (1980), p. 227-286. | MR | Zbl

[66] C. Deninger & A. Scholl - “The Beilinson conjectures” 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, 1991, p. 173-209. | MR | Zbl

[67] M. Deuring - “Die Zetafunktion einer algebraischen Kurve vom Geschlecte Eins, I-IV”, Gött. Nac. (1953-1957). | MR | Zbl

[68] B. Edixhoven - “Rational elliptic curves are modular (after Breuil, Conrad, Diamond and Taylor)”, in Sém. Bourbaki 1999/2000, Astérisque, vol. 276, Société Mathématique de France, 2002, exp. no 871, p. 161-188. | Numdam | MR | Zbl

[69] M. Eichler - “Quaternäre quadratische Formen und die Riemannsche Vermutung für die Konguenzzetafunktion”, Archiv der Mat. 5 (1954), p. 355-366. | MR | Zbl

[70] G. Faltings - “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math. 73 (1983), p. 349-366. | MR | Zbl

[71] -, “Almost étale extensions” 2002, p. 185-270. | Zbl

[72] J.-M. Fontaine - “Sur certains types de représentations p-adiques du groupe de Galois d’un corps local ; construction d’un anneau de Barsotti-Tate”, Ann. of Math. 115 (1982), p. 529-577. | MR | Zbl

[73] -, “Représentations p-adiques des corps locaux”, in The Grothendieck Festschrift, vol. 2, Progress in Math., vol. 87, Birkäuser, 1991, p. 249-309. | Zbl

[74] -, “Valeurs spéciales de fonctions L des motifs”, in Sém. Bourbaki 1991/92, Astérisque, vol. 206, Société Mathématique de France, 1992, exp. no 751, p. 205-249. | Numdam | Zbl

[75] -, “Le corps des périodes p-adiques”, in Périodes p-adiques, Astérisque, vol. 223, Société Mathématique de France, 1994, exposé II, p. 59-102. | Numdam | Zbl

[76] J.-M. Fontaine & B. Perrin-Riou - “Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions L, in Motives (Seattle), part 1, Proc. Symp. Pure Math., vol. 55, 1994, p. 599-706. | MR | Zbl

[77] T. Fukaya - “The theory of Coleman power series for K 2 , J. Algebraic Geom. 12 (2003), p. 1-80. | MR | Zbl

[78] -, “Coleman power series for K-groups and explicit reciprocity laws”, preprint, 2003. | MR

[79] M. Gealy - “Special values of p-adic L-functions associated to modular forms”, preprint, 2003.

[80] E. Ghate & V. Vatsal - “On the local behaviour of Λ-adic representations”, preprint, 2003.

[81] D. Goldfeld - “The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), p. 624-663. | Numdam | MR | Zbl

[82] R. Greenberg - “On the Birch and Swinnerton-Dyer conjecture”, Invent. Math. 72 (1983), p. 241-265. | MR | Zbl

[83] R. Greenberg & G. Stevens - p-adic L-functions and p-adic periods of modular forms”, Invent. Math. 111 (1993), p. 407-447. | MR | Zbl

[84] B. Gross & D. Zagier - “Heegner points and derivatives of L-series”, Invent. Math. 84 (1986), p. 225-320. | MR | Zbl

[85] L. Guo - “General Selmer groups and critical values of Hecke L-functions”, Math. Ann. 297 (1993), p. 221-233. | MR | Zbl

[86] L. Herr - “Sur la cohomologie galoisienne des corps p-adiques”, Bull. Soc. math. France 126 (1998), p. 563-600. | Numdam | MR | Zbl

[87] H. Hida - “Anticyclotomic Main Conjectures”, preprint, 2003. | MR | Zbl

[88] H. Hida & J. Tilouine - “Anti-cyclotomic Katz p-adic L-functions and congruence modules”, Ann. scient. Éc. Norm. Sup. 4 e série 26 (1993), p. 189-259. | Numdam | MR | Zbl

[89] -, “On the anticyclotomic main conjecture for CM fields”, Invent. Math. 117 (1994), p. 89-147. | MR | Zbl

[90] O. Hyodo - “On the Hodge-Tate decomposition in the imperfect residue field case”, J. reine angew. Math. 365 (1986), p. 97-113. | MR | Zbl

[91] A. Iovita & R. Pollack - “Iwasawa theory of Elliptic Curves at Supersingular Primes over 𝐙 p -extensions of number fields”, in MSRI Proceedings of a Conference on “Rankin's method in arithmetic”, à paraître. | Zbl

[92] A. Iovita & M. Spiess - “Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms”, Invent. Math. 154 (2003), p. 333-384. | MR | Zbl

[93] A. Iovita & A. Werner - p-adic height pairings on abelian varieties with semistable ordinary reduction”, J. reine angew. Math. 564 (2003), p. 181-203. | MR | Zbl

[94] K. Iwasawa - “On explicit formulas for the norm residue symbol”, J. Math. Soc. Japan 20 (1968), p. 151-164. | MR | Zbl

[95] H. Jacquet & J. Shalika - “A non vanishing theorem for zeta functions of 𝐆𝐋 n , Invent. Math. 38 (1976), p. 1-16. | MR | Zbl

[96] K. Kato - “The explicit reciprocity law and the cohomology of Fontaine-Messing”, Bull. Soc. math. France 119 (1991), p. 397-441. | Numdam | MR | Zbl

[97] -, “Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via 𝐁 dR , in Arithmetic Algebraic Geometry, Lect. Notes in Math., vol. 1553, Springer, 1993. | MR | Zbl

[98] -, “Euler systems, Iwasawa theory and Selmer groups”, Kodai Math. J. 22 (1999), p. 313-372. | MR | Zbl

[99] -, “Generalized explicit reciprocity laws”, in Algebraic number theory (Hapcheon/Saga, 1996), Adv. Stud. Contemp. Math. (Pusan), vol. 1, 1999, p. 57-126. | MR | Zbl

[100] -, “Hodge theory and values of zeta functions of modular forms” | Numdam | Zbl

[101] K. Kato, M. Kurihara & T. Tsuji - “Local Iwasawa theory of Perrin-Riou and syntomic complexes”, preprint, 1996.

[102] -, Cours au centre Émile Borel, premier semestre 1997.

[103] M. Kisin - “Overconvergent modular forms and the Fontaine-Mazur conjecture”, Invent. Math. 153 (2003), p. 373-454. | MR | Zbl

[104] S. Kobayashi - “Iwasawa theory for elliptic curves at supersingular primes”, Invent. Math. 152 (2003), p. 1-36. | MR | Zbl

[105] V. Kolyvagin - “Euler systems”, in The Grothendieck Festschrift, vol. 2, Progress in Math., vol. 87, Birkhäuser, 1990, p. 436-483. | MR | Zbl

[106] D. Kubert & S. Lang - “Units in the modular function fields II”, Math. Ann. 218 (1975), p. 175-189. | MR | Zbl

[107] M. Kurihara - “On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I”, Invent. Math. 149 (2002), p. 195-224. | MR | Zbl

[108] S. Lang - “Sur la conjecture de Birch-Swinnerton-Dyer (d'après J. Coates et A. Wiles)”, in Sém. Bourbaki 1976/77, Lect. Notes in Math., vol. 677, Springer, 1978, exp. no 503, p. 189-200. | Numdam | MR | Zbl

[109] -, “Les formes bilinéaires de Néron et Tate”, in Sém. Bourbaki 1963/64, Société Mathématique de France, 1995, exp. no 274, rééd. Sém Bourbaki 1948-1968, vol. 8. | Numdam | Zbl

[110] Y. Manin - “Periods of cusp forms, and p-adic Hecke series”, Math. USSR-Sb. 92 (1973), p. 371-393. | MR | Zbl

[111] B. Mazur - “Rational points of abelian varieties in towers of number fields”, Invent. Math. 18 (1972), p. 183-266. | MR | Zbl

[112] -, “Modular curves and arithmetic”, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, 1984, p. 185-211. | MR

[113] -, “On monodromy invariants occuring in global arithmetic, and Fontaine's theory”, Contemp. Math. 165 (1994), p. 1-20. | MR

[114] B. Mazur & K. Rubin - “Elliptic curves and class field theory”, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, p. 185-195. | MR | Zbl

[115] -, Kolyvagin systems, Mem. Amer. Math. Soc., vol. 168, American Mathematical Society, 2004. | MR

[116] B. Mazur & P. Swinnerton-Dyer - “Arithmetic of Weil curves”, Invent. Math. 25 (1974), p. 1-61. | MR | Zbl

[117] B. Mazur & J. Tate - “Canonical height pairings via biextensions”, in Arithmetic and Geometry : Papers dedicated to I.R. Shafarevich, Progress in Math., vol. 35, Birkhäuser, 1983, p. 195-238. | MR | Zbl

[118] -, “The p-adic sigma function”, Duke Math. J. 62 (1991), p. 663-688. | MR | Zbl

[119] B. Mazur, J. Tate & J. Teitelbaum - “On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer”, Invent. Math. 84 (1986), p. 1-48. | MR | Zbl

[120] B. Mazur & J. Tilouine - “Représentations galoisiennes, différentielles de Kähler et “conjectures principales””, Publ. Math. Inst. Hautes Études Sci. 71 (1990), p. 65-103. | Numdam | MR | Zbl

[121] B. Mazur & A. Wiles - “Class fields of abelian extensions of 𝐐, Invent. Math. 76 (1984), p. 179-330. | MR | Zbl

[122] J.-F. Mestre - “Formules explicites et minorations de conducteurs de variétés algébriques”, Comp. Math. 58 (1986), p. 209-232. | Numdam | MR | Zbl

[123] J. Nekovář - “Kolyvagin's method for Chow groups of Kuga-Sato varieties”, Invent. Math. 107 (1992), p. 99-125. | MR | Zbl

[124] -, “On p-adic height pairings”, in Séminaire de théorie des nombres 1990-1991, Progress in Math., vol. 108, Birkhäuser, 1993, p. 127-202. | MR | Zbl

[125] -, “On the p-adic heights of Heegner cycles”, Math. Ann. 302 (1995), p. 609-686. | MR | Zbl

[126] -, “On the parity of ranks of Selmer groups II”, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), p. 99-104. | MR | Zbl

[127] A. Néron - “Quasi-fonctions et hauteurs sur les variétés abéliennes”, Ann. of Math. 82 (1965), p. 249-331. | MR | Zbl

[128] -, “Fonctions thêta p-adiques et hauteurs p-adiques”, in Séminaire de Théorie des Nombres, Paris 1980-1981, Progress in Math., vol. 22, Birkhäuser, 1982. | Zbl

[129] J. Oesterlé - “Nombres de classes des corps quadratiques imaginaires”, in Sém. Bourbaki 1983/84, Astérisque, vol. 121-122, Société Mathématique de France, 1985, exp. no 631, p. 309-323. | Numdam | MR | Zbl

[130] -, “Travaux de Wiles (et Taylor,...). II”, in Sém. Bourbaki 1994/95, Astérisque, vol. 237, Société Mathématique de France, 1996, exp. no 804, p. 333-355. | Numdam | MR

[131] A. Panchishkin - “A new method of constructing p-adic L-functions associated with modular forms”, Mosc. Math. J. 2 (2002), p. 313-328. | MR | Zbl

[132] -, “Two variable p-adic L functions attached to eigenfamilies of positive slope”, Invent. Math. 154 (2003), p. 551-615. | MR | Zbl

[133] B. Perrin-Riou - “Hauteurs p-adiques”, in Séminaire de Théorie des Nombres, Paris 1982-1983, Progress in Math., vol. 51, Birkhäuser, 1984. | MR | Zbl

[134] -, “Points de Heegner et dérivées de fonctions L p-adiques”, Invent. Math. 89 (1987), p. 455-510. | MR | Zbl

[135] -, “Travaux de Kolyvagin et Rubin”, in Sém. Bourbaki 1989/90, Astérisque, vol. 189-190, Société Mathématique de France, 1990, exp. no 717, p. 69-106. | Numdam | MR

[136] -, “Théorie d’Iwasawa et hauteurs p-adiques”, Invent. Math. 109 (1992), p. 137-185. | MR | Zbl

[137] -, “Fonctions L p-adiques d’une courbe elliptique et points rationnels”, Ann. Inst. Fourier (Grenoble) 43 (1993), p. 945-995. | Numdam | MR | Zbl

[138] -, “Théorie d’Iwasawa des représentations p-adiques sur un corps local”, Invent. Math. 115 (1994), p. 81-149. | MR | Zbl

[139] -, Fonctions L p-adiques des représentations p-adiques, Astérisque, vol. 229, Société Mathématique de France, 1995. | Numdam | MR | Zbl

[140] -, “Systèmes d’Euler et représentations p-adiques”, Ann. Inst. Fourier (Grenoble) 48 (1998), p. 1231-1307. | MR

[141] -, “Représentations p-adiques et normes universelles I, le cas cristallin”, J. Amer. Math. Soc. 13 (2000), p. 533-551. | MR | Zbl

[142] -, “Arithmétique des courbes elliptiques à réduction supersingulière en p, preprint, 2001. | Zbl

[143] -, Théorie d’Iwasawa des représentations p-adiques semi-stables, Mém. Soc. math. France (N.S.), vol. 84, Société Mathématique de France, 2001.

[144] -, “Quelques remarques sur la théorie d'Iwasawa des courbes elliptiques”, in Number theory for the millennium, III (Urbana, IL, 2000), 2002, p. 119-147. | Zbl

[145] R. Pollack - “On the p-adic L-function of a modular form at a supersingular prime”, Duke Math. J. 118 (2003), p. 523-558. | MR | Zbl

[146] R. Pollack & K. Rubin - “The main conjecture for CM elliptic curves at supersingular primes”, Ann. of Math. 159 (2004), p. 447-464. | MR | Zbl

[147] R. Pollack & G. Stevens - “The “missing” p-adic L-function”, preprint, 2003. | MR

[148] K. Ribet - “Galois representations attached to modular forms”, Invent. Math. 28 (1975), p. 245-275. | MR | Zbl

[149] -, “A modular construction of unramified extensions of 𝐐(ζ p ), Invent. Math. 34 (1976), p. 151-162. | MR | Zbl

[150] -, “Galois representations attached to modular forms II”, Glasgow Math. J. 27 (1985), p. 185-194. | MR | Zbl

[151] D. Rohrlich - “On L-functions of elliptic curves and cyclotomic towers”, Invent. Math. 75 (1984), p. 409-423. | MR | Zbl

[152] K. Rubin - “Elliptic curves and 𝐙 p -extensions”, Comp. Math. 56 (1985), p. 237-250. | Numdam | MR | Zbl

[153] -, “Local units, elliptic units, Heegner points, and elliptic curves”, Invent. Math. 88 (1987), p. 405-422. | MR | Zbl

[154] -, “Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication”, Invent. Math. 89 (1987), p. 527-560. | MR | Zbl

[155] -, “The “main conjectures” of Iwasawa theory for imaginary quadratic fields”, Invent. Math. 103 (1991), p. 25-68. | MR | Zbl

[156] -, Euler systems, Ann. of Math. Studies, vol. 147, Princeton Univ. Press, 2000.

[157] P. Schneider - p-adic height pairings, II”, Invent. Math. 79 (1985), p. 329-374. | MR | Zbl

[158] P. Schneider & J. Teitelbaum - p-adic Fourier theory”, Doc. Math. 6 (2001), p. 447-481. | MR | Zbl

[159] A. Scholl - “Motives for modular forms”, Invent. Math. 100 (1990), p. 419-430. | MR | Zbl

[160] -, “Remarks on special values of L-functions”, in L-functions and Arithmetic, Proc. of the Durham Symp., London Math. Soc. L.N.S., vol. 153, Cambridge University Press, 1991, p. 373-392. | MR | Zbl

[161] -, “An introduction to Kato's Euler systems”, in Galois representations in arithmetic algebraic geometry, Cambridge University Press, 1998, p. 379-460. | MR | Zbl

[162] -, “Higher regulators and special values of L-functions of modular forms”, en préparation.

[163] -, “Zeta elements for higher weight modular forms”, en préparation.

[164] J.-P. Serre - Abelian -adic representations and elliptic curves, W. A. Benjamin, 1968. | MR | Zbl

[165] -, “Formes modulaires et fonctions zêta p-adiques”, in Modular functions of one variable III, Lect. Notes in Math., vol. 350, Springer, 1972, p. 191-268. | Zbl

[166] -, “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”, Invent. Math. 15 (1972), p. 259-331. | MR | Zbl

[167] -, “Travaux de Wiles (et Taylor,...). I”, in Sém. Bourbaki 1994/95, Astérisque, vol. 237, Société Mathématique de France, 1996, exp. no 803, p. 319-332 | Numdam | MR

[168] -, lettre du 13/11/59, in Correspondance Grothendieck-Serre, Documents Mathématiques, vol. 2 Société Mathématique de France, 2001. | Zbl

[169] G. Shimura - “Correspondances modulaires et les fonctions ζ de courbes algébriques”, J. Math. Soc. Japan 10 (1958), p. 1-28. | MR | Zbl

[170] -, “Sur les intégrales attachées aux formes automorphes”, J. Math. Soc. Japan 11 (1959), p. 291-311. | MR | Zbl

[171] -, Introduction to the arithmetic theory of automorphic functions, Kan Memorial Lectures 1, vol. 11, Math. Soc. of Japan, 1971. | Zbl

[172] -, “On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields”, Nagoya Math. J. 43 (1971), p. 199-208. | MR | Zbl

[173] -, “On the factors of the jacobian variety of a modular function field”, J. Math. Soc. Japan 25 (1973), p. 523-544. | MR | Zbl

[174] -, “The special values of the zeta functions associated with cusp forms”, Comm. Pure Appl. Math. 29 (1976), p. 783-804. | MR | Zbl

[175] C. Skinner & E. Urban - “Sur les déformations p-adiques des formes de Saito-Kurokawa”, C. R. Acad. Sci. Paris Sér. I Math. 335 (2002), p. 581-586. | MR | Zbl

[176] -, en préparation.

[177] C. Soulé - “Régulateurs”, in Sém. Bourbaki 1984/85, Astérisque, vol. 133-134, Société Mathématique de France, 1986, exp. no 644, p. 237-253. | Numdam | MR | Zbl

[178] -, “Éléments cyclotomiques en K-théorie”, in Journées Arithmétiques, (Besançon, 1985), Astérisque, vol. 147-148, Société Mathématique de France, 1987, p. 225-257. | Numdam | Zbl

[179] G. Stevens - “Coleman’s -invariant and families of modular forms”, preprint, 1996.

[180] -, Cours au centre Émile Borel, premier semestre 2000.

[181] J. Tate - p-divisible groups”, in Proc. of a conference on local fields, Nuffic Summer School at Driebergen, Springer, Berlin, 1967, p. 158-183. | MR | Zbl

[182] -, “A review of non-Archimedean elliptic functions”, in Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995, p. 162-184. | MR | Zbl

[183] -, “On the conjectures of Birch and Swinnerton-Dyer and a geometric analog”, in Sém. Bourbaki 1965/66, Société Mathématique de France, 1995, exp. no 306, rééd. Sém. Bourbaki 1948-1968, vol. 9. | Numdam | MR | Zbl

[184] F. Thaine - “On the ideal class group of real abelian number fields”, Ann. of Math. 128 (1988), p. 1-18. | MR | Zbl

[185] J. Tilouine - “Sur la conjecture principale anticyclotomique”, Duke Math. J. 59 (1989), p. 629-673. | MR | Zbl

[186] T. Tsuji - p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case”, Invent. Math. 137 (1999), p. 233-411. | MR | Zbl

[187] -, “Semi-stable conjecture of Fontaine-Jannsen : a survey” 2002, p. 323-370. | Numdam | MR | Zbl

[188] V. Vatsal - “Uniform distribution of Heegner points”, Invent. Math. 148 (2002), p. 1-46. | MR | Zbl

[189] M. Vishik - “Non-archimedian measures connected with Dirichlet series”, Math. USSR-Sb. 28 (1976), p. 216-228. | Zbl

[190] M. Waldschmidt - “Sur la nature arithmétique des valeurs de fonctions modulaires”, in Sém. Bourbaki 1996/97, Astérisque, vol. 245, Société Mathématique de France, 1997, exp. no 824, p. 105-140. | Numdam | MR | Zbl

[191] A. Weil - Elliptic functions according to Eisenstein and Kronecker, Erg. der Math., vol. 88, Springer-Verlag, 1976. | MR | Zbl

[192] A. Wiles - “Higher explicit reciprocity laws”, Ann. of Math. 107 (1978), p. 235-254. | MR | Zbl

[193] -, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. 141 (1995), p. 443-551. | MR | Zbl

[194] Y. Zarhin - p-adic heights on abelian varieties”, in Séminaire de Théorie des Nombres, Paris 1987-1988, Progress in Math., vol. 81, Birkhäuser, 1989. | MR | Zbl